
Since \(|h_l|\)  and \(|\text{g}_l|\)  are assumed to be independently Rayleigh distributed random variables (RVs), the mean value and the variance of their product are \(\mathbf {E}\left[|h_l||\text{g}_l|\right]=0.25\pi \)  and \(\mathbf {Var}\left[|h_l||\text{g}_l|\right]=1-0.0625\pi ^2\) , respectively, where \(\mathbf {E}\left[\cdot \right]\)  is an expectation operator and \(\mathbf {Var}\left[|h_l||\text{g}_l|\right]\)  is a variance operator. For a sufficiently large value of \(N\)  i.e., \(N>>1\) , according to the central limit theorem, \(\sum _{l=1}^{N}|h_l||\text{g}_l|\)  converges to a Gaussian distributed RV with statistical parameters \(\mathbf {E}\left[\sum _{l=1}^{N}|h_l||\text{g}_l|\right]=0.25\pi N\)  and \(\mathbf {Var}\left[\sum _{l=1}^{N}|h_l||\text{g}_l|\right]=\left(1-0.0625\pi ^2\right)N\) . Hence,
the RV \(\gamma \)  is a non-central chi-square RV with one degree of freedom and has the following
moment generating function (MGF) \(\mathcal {G}_{\gamma }(s)\)  [1]}, [2]}
\(\mathcal {G}_{\gamma }(s)&\approx \left(\frac{\Delta _1}{\Delta _1+s\bar{\gamma }}\right)^{0.5}\,\exp \left(-\frac{s\bar{\gamma }\Delta _2}{\Delta _1+s\bar{\gamma }}\right),\) 
